According to my book the spring constant is given by k = V ″ ( X o ). Where V ( X ) is the potential energy function. if I use the function V ( X ) = X 6 + X 4 the spring constant is zero at X o = 0. However, the plot looks like a stable equilibrium that would allow harmonic oscillation near X o . So why is k zero in this case?

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According to my book the spring constant is given by   k  =      V    ″    (      X    o    ). Where   V  (  X  ) is the potential energy function. if I use the function   V  (  X  )  =      X    6    +      X    4   the spring constant is zero at       X    o    =  0. However, the plot looks like a stable equilibrium that would allow harmonic oscillation near       X    o  . So why is k zero in this case?

Krha77 · Accepted Answer

The force is given by the first derivative of the potential:  F  (  X  )  =  −      d          d      X        V  (  X  )  =  −  6      X    5    −  4      X    3    .At small displacements, i.e. when   X  ≪  1 we have       X    5    ≪      X    3  , that is the returning force is proportional to       X    3   and not to X, as is in the case of a harmonic oscillator. The spring constant is indeed zero.You are right that there are oscillations in this case, however these oscillations are not harmonic. Your logic based on the shape of the potential energy is a sound one and is used for analyzing nonlinear oscillators, such as the one that you have here.

samuelaplc · Accepted Answer

Periodic motion is not the same as being a harmonic oscillator. A harmonic oscillator describes the very particular force configuration F=-kx, where k is the spring constant, and this is not the force for the potential you propose. Therefore, there need not be an equivalent of a spring constant just because the motion is periodic. For example, a particle bouncing around an infinite square potential well would exhibit periodic motion, but not oscillation.

According to my book the spring constant is given by k=V′′(X_o).

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